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Let $\mathfrak{C}$ be the smallest class of countable discrete groups withthe following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii)$\mathfrak{C}$ is closed under isomorphisms, countable increasing unions andextensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countablediscrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also,$\mathfrak{C}$ is a subclass of the class of countable discrete torsion-freeelementary amenable groups. In this paper, we show that if $\Gamma\in\mathfrak{C}$, then all strongly outer actions of $\Gamma$ on the Razak-Jacelonalgebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regardedas an analogous result of Szab\'o's result for strongly self-absorbingC$^*$-algebras.

Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra